Polynomial Relations for q-Characters via the ODE/IM Correspondence

Let $U_q(\mathfrak{b})$ be the Borel subalgebra of a quantum affine algebra of type $X^{(1)}_n$ ($X=A,B,C,D$). Guided by the ODE/IM correspondence in quantum integrable models, we propose conjectural polynomial relations among the $q$-characters of certain representations of $U_q(\mathfrak{b})$

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Does Format of Pricing Contract Matter?

The use of linear wholesale price contract has long been recognized as a threat to achieving channel effciency. Many formats of nonlinear pricing contract have been proposed to achieve vertical channel coordination. Examples include two-part tariff and quantity discount. A two-part tariff charges the downstream party a fixed fee for participation and a uniform unit price. A quantity discount contract does not include a fixed fee and charges a lower unit price for each additional unit. Extant economic theories predict these contracts, when chosen optimally, to be revenue and division equivalent in that they all restore full channel effciency and give the same surplus to the upstream party assuming constant relative bargaining power. We conduct a laboratory experiment to test the empirical equivalence of the two pricing formats. Surprisingly, both pricing formats fail to coordinate the channel even in a well-controlled market environment with subjects motivated by significant monetary incentives. The observed channl effciencies were significantly lower than 100%. In fact, they are statistically no better than that of the linear wholesale price contract. Revenue equivalence fails because the quantity discount scheme achieves a higher channel effciency than the two-part tariff. Also, division equivalence does not hold because the quantity discount scheme accords a higher surplus to the upstream party than the two-part tariff. To account for the observed empirical regularities, we allow the downstream party to have a reference-dependent utility in which the upfront fixed fee is framed as loss andn the subsequent contribution margin as gain. The proposed model nests the standard economic model as a special case with a loss aversion coeffcient of 1.0. The estimated loss aversion coeffcient is 1.6, thereby rejecting the standard model. We rule out other plausible explanations such as parties having fairness concerns and non-linear risk attitudes.Pricing Format, Two-Part Tariff, Quantity Discount, Channel Efficiency, Double Marginalization, Reference-Dependent Utility, Experimental Economics, Behavioral Economics